Those are all very good signs. They all show that she is still very much interested in your life. As for your question on inviting her to a day with the boys, I think that is OK. I would suggest asking it in such a way as "I am going with the boys to ???, you are welcome to come along if you like." However, I would not do this too much. Anything more and it would look like pursuing. When she is ready, she will start inviting you to things as well.
